1. $C$ is well-defined since $I(X;Y)=I(P_{X},P_{Y|X})$ is a concave and continuous function of $p_{X}$ (by concavity of information measure) and the set of $p_{X}$ is compact (closed and bounded) in $\mathbb{R}^{|\mathcal{X}|}$; thus $I(X;Y)$ admits a maximum. Also since $$0\le I(X;Y)\le \min\{\log|\mathcal{X}|,\log|\mathcal{Y}|\},$$then $$0\le C\le \min\{\log|\mathcal{X}|,\log|\mathcal{Y}|\}$$ 2. Will see via Shannon’s Channel Coding Theorem for the DMC that $C$ is the DMC’s “operational capacity”